A Passion for Mathematics: Numbers, Puzzles, Madness, Religion, and the Quest for Reality

Table of Contents

Works by Clifford A. Pickover

Title Page

Copyright Page

Epigraph

Acknowledgments

Introduction

The Ramanujan Code

Blood Dreams and God’s Mathematicians

The Mathematical Smorgasbord

Explanation of Symbols

Cultivating Perpetual Mystery

Chapter 1 - Numbers, History, Society, and People

Chapter 2 - Cool Numbers

Chapter 3 - Algebra, Percentages, Weird Puzzles, and Marvelous Mathematical Manipulations

Chapter 4 - Geometry, Games, and Beyond

Chapter 5 - Probability:Take Your Chances

Chapter 6 - Big Numbers and Infinity

Chapter 7 - Mathematics and Beauty

Answers

References

Index

About the Author

Works by Clifford A. Pickover

The Alien IQ Test

Black Holes: A Traveler’s Guide

Calculus and Pizza

Chaos and Fractals

Chaos in Wonderland

Computers, Pattern, Chaos, and Beauty

Computers and the Imagination

Cryptorunes: Codes and Secret Writing

Dreaming the Future

Egg Drop Soup

Future Health

Fractal Horizons: The Future Use of Fractals

Frontiers of Scientific Visualization

The Girl Who Gave Birth to Rabbits

Keys to Infinity

Liquid Earth

The Lobotomy Club

The Loom of God

The Mathematics of Oz

Mazes for the Mind: Computers and the Unexpected

Mind-Bending Visual Puzzles (calendars and card sets)

The Paradox of God and the Science of Omniscience

The Pattern Book: Fractals, Art, and Nature

The Science of Aliens

Sex, Drugs, Einstein, and Elves

Spider Legs (with Piers Anthony)

Spiral Symmetry (with Istvan Hargittai)

Strange Brains and Genius

Sushi Never Sleeps

The Stars of Heaven

Surfing through Hyperspace

Time: A Traveler’s Guide

Visions of the Future

Visualizing Biological Information

Wonders of Numbers

The Zen of Magic Squares, Circles, and Stars

Copyright © 2005 by Clifford A. Pickover. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

Illustration credits: pages 91, 116, 137, 140, 142, 149, 150, 151, 157, 158, 159, 160, 162, 164, 167, 168, 169, 179, 199, 214, 215, 224, 225, 230, 274, 302, 336, 338, 341, 343, 345, and 348 by Brian C. Mansfield; 113, 114, 115, 145, 146, 332, 333, and 334 by Sam Loyd; 139 courtesy of Peter Hamburger and Edit Hepp; 141 by Stewart Raphael, Audrey Raphael, and Richard King; 155 by Patrick Grimm and Paul St. Denis; 165 and 166 from Magic Squares and Cubes by W. S. Andrews; 177 and 352 by Henry Ernest Dudeney; 200 by Bruce Patterson; 204 by Bruce Rawles; 206 by Jürgen Schmidhuber; 253 by Abram Hindle; 254, 255, 256, and 257 by Chris Coyne; 258 and 259 by Jock Cooper; 260 by Linda Bucklin; 261 by Sally Hunter; 262 and 263 by Jos Leys; 264 by Robert A. Johnston; and 266 by Cory and Catska Ench.

Design and composition by Navta Associates, Inc.

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Library of Congress Cataloging-in-Publication Data:

Pickover, Clifford A.

A passion for mathematics : numbers, puzzles, madness, religion, and the quest for reality / Clifford A. Pickover.

p. cm.

Includes bibliographical references and index.

ISBN-13 978-0-471-69098-6 (paper)

ISBN-10 0-471-69098-8 (paper)

1. Mathematics. I. Title

QA39.3.P53 2005

510—dc22

2004060622

Ramanujan said that he received his formulas from God. This book is dedicated to all those who find Ramanujan’s π formulas beautiful to look at:

(where Poch(n) refers to the Pochhammer notation described in chapter 2)

Truly the gods have not from the beginning revealed all things to mortals, but by long seeking, mortals make progress in discovery.

—Xenophanes of Colophon (c. 500 B.C.)

Every blade of grass has its angel that bends over it and whispers, ‘grow, grow.’

—Talmudic commentary Midrash Bereishis Rabbah, 10:6

Acknowledgments

I thank Brian Mansfield for his wonderful cartoon diagrams that appear throughout the book. Over the years, Brian has been helpful beyond compare.

Numerous people have provided useful feedback and information relating to the solutions to my puzzles; these individuals include Dennis Gordon, Robert Stong, Paul Moskowitz, Joseph Pe, Daniel Dockery, Mark Nandor, Mark Ganson, Nick Hobson, Chuck Gaydos, Graham Cleverley, Jeffrey Carr, Jon Anderson, Jaymz James Salter, Chris Meyers, Pete Barnes, Steve Brazzell, Steve Blattnig, Edith Rudy, Eric Baise, Martie Saxenmeyer, Bob Ewell, Teja Krasek, and many more. I discussed some of the original puzzles in this book at my Pickover Discussion Group, located on the Web at groups.yahoo.com/group/CliffordPickover, and I thank the group members for their wonderful discussions and comments.

Many of the fancy formulas derived by the Indian mathematician Srinivasa Ramanujan come from Calvin Clawson’s Mathematical Mysteries, Bruce Berndt’s Ramanujan’s Notebooks (volumes 1 and 2), and various Internet sources. With respect to some of the elegant prime number formulas, Clawson cites Paulo Ribenboim’s The Book of Prime Number Records, second edition (New York: Springer, 1989), and The Little Book of Big Primes (New York: Springer, 1991). Calvin Clawson’s Mathematical Mysteries and David Wells’s various Penguin dictionaries provide a gold mine of mathematical concepts. A few problems in this book draw on, update, or revise problems in my earlier books, with information provided by countless readers who spend their lives tirelessly tackling mathematical conundrums.

Other interesting sources and recommended reading are given in the reference section. Numerous Web sites are proliferating that give comprehensive mathematical information, and my favorites include Wikipedia, the Free Encyclopedia (wikipedia.com), Ask Dr. Math (mathforum.org/dr.math/), The MacTutor History of Mathematics Archive (www.gap-system.org/~history/), The Published Data of Robert Munafo (www.mrob.com/pub/index.html), and MathWorld ( mathworld.wolfram.com).

Introduction

An equation means nothing to me unless it expresses a thought of God.

—Srinivasa Ramanujan (1887-1920)

The Ramanujan Code

An intelligent observer seeing mathematicians at work might conclude that they are devotees of exotic sects, pursuers of esoteric keys to the universe.

—Philip Davis and Reuben Hersh, The Mathematical Experience, 1981

Readers of my popular mathematics books already know how I feel about numbers. Numbers are portals to other universes. Numbers help us glimpse a greater universe that’s normally shielded from our small brains, which have not evolved enough to fully comprehend the mathematical fabric of the universe. Higher mathematical discussions are a little like poetry. The Danish physicist Niels Bohr felt similarly about physics when he said, We must be clear that, when it comes to atoms, language can be used only as in poetry.

When I think about the vast ocean of numbers that humans have scooped from the shoreless sea of reality, I get a little shiver. I hope you’ll shiver, too, as you glimpse numbers that range from integers, fractions, and radicals to stranger beasts like transcendental numbers, transfinite numbers, hyperreal numbers, surreal numbers, nimbers, quaternions, biquaternions, sedenions, and octonions. Of course, we have a hard time thinking of such queer entities, but from time to time, God places in our midst visionaries who function like the biblical prophets, those individuals who touched a universe inches away that most of us can barely perceive.

Srinivasa Ramanujan was such a prophet. He plucked mathematical ideas from the ether, out of his dreams. Ramanujan was one of India’s greatest mathematical geniuses, and he believed that the gods gave him insights. These came in a flash. He could read the codes in the mathematical matrix in the same way that Neo, the lead character in the movie The Matrix, could access mathematical symbols that formed the infrastructure of reality as they cascaded about him. I don’t know if God is a cryptographer, but codes are all around us waiting to be deciphered. Some may take a thousand years for us to understand. Some may always be shrouded in mystery.

In The Matrix Reloaded, the wise Architect tells Neo that his life is the sum of a remainder of an unbalanced equation inherent in the programming of the matrix. Similarly, the great Swiss architect Le Corbusier (1887-1965) thought that gods played with numbers in a matrix beyond our ordinary reality:

The chamois making a gigantic leap from rock to rock and alighting, with its full weight, on hooves supported by an ankle two centimeters in diameter: that is challenge and that is mathematics. The mathematical phenomenon always develops out of simple arithmetic, so useful in everyday life, out of numbers, those weapons of the gods: the gods are there, behind the wall, at play with numbers. (Le Corbusier, The Modulor, 1968)

A century ago, Ramanujan was The Matrix’s Neo in our own reality. As a boy, Ramanujan was slow to learn to speak. He seemed to spend all of his time scribbling strange symbols on his slate board or writing equations in his personal notebooks. Later in life, while working in the Accounts Department of the Port Trust Office at Madras, he mailed some of his equations to the renowned British mathematician G. H. Hardy. Hardy immediately tossed these equations into the garbage—but later retrieved them for a second look. Of the formulas, Hardy said that he had never seen anything in the least like them before, and that some had completely defeated him. He quickly realized that the equations could only be written down by a mathematician of the highest class. Hardy wrote in Ramanujan: Twelve Lectures that the formulas must be true because, if they were not true, no one would have had the imagination to invent them.

Indeed, Ramanujan often stated a result that had come from some sense of intuition out of the unconscious realm. He said that an Indian goddess inspired him in his dreams. Not all of his formulas were perfect, but the avalanche of actual gems that he plucked from the mine of reality continues to boggle our modern minds. Ramanujan said that only in mathematics could one have a concrete realization of God.

Blood Dreams and God’s Mathematicians

Repeatedly, [Western mathematicians] have been reduced to inchoate expressions of wonder and awe in the face of Ramanujan’s powers—have stumbled about, groping for words, in trying to convey the mystery of Ramanujan."

—Robert Kanigel, The Man Who Knew Infinity, 1991

According to Ramanujan, the gods left drops of vivid blood in his dreams. After he saw the blood, scrolls containing complicated mathematics unfolded before him. When Ramanujan awakened in the morning, he scribbled only a fraction of what the gods had revealed to him.

In The Man Who Knew Infinity, Robert Kanigel suggests that the ease with which Ramanujan’s spirituality and mathematics intertwined signified a peculiar flexibility of mind, a special receptivity to loose conceptual linkages and tenuous associations.... Indeed, Ramanujan’s openness to mystical visions suggested a mind endowed with slippery, flexible, and elastic notions of cause and effect that left him receptive to what those equipped with purely logical gifts could not see.

Before we leave Ramanujan, I should point out that many other mathematicians, such as Carl Friedrich Gauss, James Hopwood Jeans, Georg Cantor, Blaise Pascal, and John Littlewood, believed that inspiration had a divine aspect. Gauss said that he once proved a theorem not by dint of painful effort but so to speak by the grace of God. For these reasons, I have included a number of brief pointers to religious mathematicians in chapter 1. I hope these examples dispel the notion that mathematics and religion are totally separate realms of human endeavor.

Our mathematical description of the universe forever grows, but our brains and language skills remain entrenched. New kinds of mathematics are being discovered or created all the time, but we need fresh ways to think and to understand. For example, in the last few years, mathematical proofs have been offered for famous problems in the history of mathematics, but the arguments have been far too long and complicated for experts to be certain they are correct. The mathematician Thomas Hales had to wait five years before expert reviewers of his geometry paper—submitted to the journal Annals of Mathematics—finally decided that they could find no errors and that the journal should publish Hale’s proof, but only with a disclaimer saying they were not certain it was right! Moreover, mathematicians such as Keith Devlin have admitted (in the May 25, 2004, New York Times) that the story of mathematics has reached a stage of such abstraction that many of its frontier problems cannot even be understood by the experts. There is absolutely no hope of explaining these concepts to a popular audience. We can construct theories and do computations, but we may not be sufficiently smart to comprehend, explain, or communicate these ideas.

A physics analogy is relevant here. When Werner Heisenberg worried that human beings might never truly understand atoms, Bohr was a bit more optimistic. He replied, "I think we may yet be able to do so, but in the process we may have to learn what the word understanding really means." Today, we use computers to help us reason beyond the limitations of our own intuition. In fact, experiments with computers are leading mathematicians to discoveries and insights never dreamed of before the ubiquity of these devices. Computers and computer graphics allow mathematicians to discover results long before they can prove them formally, thus opening entirely new fields of mathematics. Even simple computer tools, such as spreadsheets, give modern mathematicians power that Heisenberg, Einstein, and Newton would have lusted after. As just one example, in the late 1990s, computer programs designed by David Bailey and Helaman Ferguson helped to produce new formulas that related pi to log 5 and two other constants. As Erica Klarreich reports in the April 24, 2004, edition of Science News, once the computer had produced the formula, proving that it was correct was extremely easy. Often, simply knowing the answer is the largest hurdle to overcome when formulating a proof.

The Mathematical Smorgasbord

As the island of knowledge grows, the surface that makes contact with mystery expands. When major theories are overturned, what we thought was certain knowledge gives way, and knowledge touches upon mystery differently. This newly uncovered mystery may be humbling and unsettling, but it is the cost of truth. Creative scientists, philosophers, and poets thrive at this shoreline.

—W. Mark Richardson, A Skeptic’s Sense of Wonder, Science, 1998

Despite all of my mystical talk about mathematics and the divine, mathematics is obviously practical. Mathematics has affected virtually every field of scientific endeavor and plays an invaluable role in fields ranging from science to sociology, from modeling ecological disasters and the spread of diseases to understanding the architecture of our brains. Thus, the fun and quirky facts, questions, anecdotes, equations, and puzzles in this book are metaphors for an amazing range of mathematical applications and notations. In fact, this book is a smorgasbord of puzzles, factoids, trivia, quotations, and serious problems to consider. You can pick and choose from the various delicacies as you explore the platter that’s set before you. The problems vary in scope, so you are free to browse quickly among concepts ranging from Champernowne’s number to the Göbel number, a number so big that it makes a trillion pale in comparison. Some of the puzzles are arranged randomly to enhance the sense of adventure and surprise. My brain is a runaway train, and these puzzles and factoids are the chunks of cerebrum scattered on the tracks.

Occasionally, some of the puzzles in this book will seem simple or frivolous; for example, Why does a circle have 360 degrees? Or, Is zero an even number? Or, What’s the hardest license plate to remember? Or, Could Jesus calculate 30 × 24? However, these are questions that fans often pose to me, and I love some of these quirkies the best. I agree with the Austrian physicist Paul Ehrenfest, who said, Ask questions. Don’t be afraid to appear stupid. The stupid questions are usually the best and hardest to answer. They force the speaker to think about the basic problem.

In contrast to the quirkies, some of the puzzles I pose in this book are so insanely difficult or require such an exhaustive search that only a computer hacker could hope to answer them, such as my problem Triangle of the Gods (see page 61).

These are questions with which I have challenged my geeky colleagues, and on which they have labored for hours and sometimes days. I think you’ll enjoy seeing the results. Don’t be scared if you have no chance in hell of solving them. Just enjoy the fact that intense people will often respond to my odd challenges with apparent glee. Most of the problems in the book are somewhere in-between the extremes of simplicity and impossibility and can be solved with a pencil and paper. Chapter 3 contains most of the problems that teachers can enjoy with students.

I will also tease you with fancy formulas, like those decorating the book from Ramanujan. Sometimes my goal is simply to delight you with wonderful-looking equations to ponder. Occasionally, a concept is repeated, just to see if you’ve learned your lesson and recognize a similar problem in a new guise. The different ways of getting to the same solution or concept reveal things that a single approach misses.

I’ve been in love with recreational mathematics for many years because of its educational value. Contemplating even simple problems stretches the imagination. The usefulness of mathematics allows us to build spaceships and investigate the geometry of our universe. Numbers will be our first means of communication with intelligent alien races.

Ancient peoples, like the Greeks, also had a deep fascination with numbers. Could it be that in difficult times numbers were the only constant thing in an ever-shifting world? To the Pythagoreans, an ancient Greek sect, numbers were tangible, immutable, comfortable, eternal—more reliable than friends, less threatening than Zeus.

Explanation of Symbols

Non-Euclidean calculus and quantum physics are enough to stretch any brain, and when one mixes them with folklore, and tries to trace a strange background of multidimensional reality behind the ghoulish hints of the Gothic tales and the wild whispers of the chimney-corner, one can hardly expect to be wholly free from mental tension.

—H. P. Lovecraft, Dreams in the Witch House, 1933

I use the following symbols to differentiate classes of entries in this book:

003 signifies a thought-provoking quotation.

004 signifies a mathematical definition that may come in handy throughout the book.

005 signifies a mathematical factoid to stimulate your imagination.

006 signifies a problem to be solved. Answers are provided at the back of the book.

These different classes of entries should cause even the most rightbrained readers to fall in love with mathematics. Some of the zanier problems will entertain people at all levels of mathematical sophistication. As I said, don’t worry if you cannot solve many of the puzzles in the book. Some of them still challenge seasoned mathematicians.

One common characteristic of mathematicians is an obsession with completeness—an urge to go back to first principles to explain their works. As a result, readers must often wade through pages of background before getting to the essential ingredients. To avoid this, each problem in my book is short, at most only a few paragraphs in length. One advantage of this format is that you can jump right in to experiment or ponder and have fun, without having to sort through a lot of verbiage. The book is not intended for mathematicians looking for formal mathematical explanations. Of course, this approach has some disadvantages. In just a paragraph or two, I can’t go into any depth on a subject. You won’t find much historical context or many extended discussions. In the interest of brevity, even the answer section may require readers to research or ponder a particular puzzle further to truly understand it.

To some extent, the choice of topics for inclusion in this book is arbitrary, although these topics give a nice introduction to some classic and original problems in number theory, algebra, geometry, probability, infinity, and so forth. These are also problems that I have personally enjoyed and are representative of a wider class of problems of interest to mathematicians today. Grab a pencil. Do not fear. Some of the topics in the book may appear to be curiosities, with little practical application or purpose. However, I have found these experiments to be useful and educational, as have the many students, educators, and scientists who have written to me. Throughout history, experiments, ideas, and conclusions that originate in the play of the mind have found striking and unexpected practical applications.

A few puzzles come from Sam Loyd, the famous nineteenth-century American puzzlemaster. Loyd (1841-1911) invented thousands of popular puzzles, which his son collected in a book titled Cyclopedia of Puzzles. I hope you enjoy the classics presented here.

Cultivating Perpetual Mystery

Pure mathematics is religion.

—Friedrich von Hardenberg, circa 1801

A wonderful panoply of relationships in nature can be expressed using integer numbers and their ratios. Simple numerical patterns describe spiral floret formations in sunflowers, scales on pinecones, branching patterns on trees, and the periodic life cycles of insect populations. Mathematical theories have predicted phenomena that were not confirmed until years later. Maxwell’s equations, for example, predicted radio waves. Einstein’s field equations suggested that gravity would bend light and that the universe is expanding. The physicist Paul Dirac once noted that the abstract mathematics we study now gives us a glimpse of physics in the future. In fact, his equations predicted the existence of antimatter, which was subsequently discovered. Similarly, the mathematician Nikolai Lobachevsky said that there is no branch of mathematics, however abstract, which may not someday be applied to the phenomena of the real world.

A famous incident involving Murray Gell-Mann and his colleagues demonstrates the predictive power of mathematics and symmetry regarding the existence of a subatomic particle known as the Omega-minus. Gell-Mann had drawn a symmetric, geometric pattern in which each position in the pattern, except for one empty spot, contained a known particle. Gell-Mann put his finger on the spot and said with almost mystical insight, There is a particle. His insight was correct, and experimentalists later found an actual particle corresponding to the empty spot.

One of my favorite quotations describing the mystical side of science comes from Richard Power’s The Gold Bug Variations: Science is not about control. It is about cultivating a perpetual condition of wonder in the face of something that forever grows one step richer and subtler than our latest theory about it. It is about reverence, not mastery.

Today, mathematics has permeated every field of scientific endeavor and plays an invaluable role in biology, physics, chemistry, economics, sociology, and engineering. Math can be used to help explain the structure of a rainbow, teach us how to make money in the stock market, guide a spacecraft, make weather forecasts, predict population growth, design buildings, quantify happiness, and analyze the spread of AIDS.

Mathematics has caused a revolution. It has shaped our thoughts. It has shaped the way we think.

Mathematics has changed the way we look at the world.

This introduction is dedicated to anyone who can decode the following secret message.

I am the thought you are now thinking.

—Douglas Hofstadter, Metamagical Themas, 1985

1

Numbers, History, Society, and People

IN WHICH WE ENCOUNTER RELIGIOUS MATHEMATICIANS, MAD MATHEMATICIANS, famous mathematicians, mathematical savants, quirky questions, fun trivia, brief biographies, mathematical gods, historical oddities, numbers and society, gossip, the history of mathematical notation, the genesis of numbers, and What if? questions.

Mathematics is the hammer that shatters the ice of our unconscious.

008 Ancient counting. Let’s start the book with a question. What is the earliest evidence we have of humans counting? If this question is too difficult, can you guess whether the evidence is before or after 10,000 B.C.—and what the evidence might be? (See Answer 1.1.)

009 Mathematics and beauty. I’ve collected mathematical quotations since my teenage years. Here’s a favorite: Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture (Bertrand Russell, Mysticism and Logic, 1918).

010 The symbols of mathematics. Mathematical notation shapes humanity’s ability to efficiently contemplate mathematics. Here’s a cool factoid for you: The symbols + and -, referring to addition and subtraction, first appeared in 1456 in an unpublished manuscript by the mathematician Johann Regiomontanus (a.k.a. Johann Müller). The plus symbol, as an abbreviation for the Latin et (and), was found earlier in a manuscript dated 1417; however, the downward stroke was not quite vertical.

011 Mathematics and reality. Do humans invent mathematics or discover mathematics? (See Answer 1.2.)

012 Mathematics and the universe. Here is a deep thought to start our mathematical journey. Do you think humanity’s long-term fascination with mathematics has arisen because the universe is constructed from a mathematical fabric? We’ll approach this question later in the chapter. For now, you may enjoy knowing that in 1623, Galileo Galilei echoed this belief in a mathematical universe by stating his credo: Nature’s great book is written in mathematical symbols. Plato’s doctrine was that God is a geometer, and Sir James Jeans believed that God experimented with arithmetic. Isaac Newton supposed that the planets were originally thrown into orbit by God, but even after God decreed the law of gravitation, the planets required continual adjustments to their orbits.

013 Math beyond humanity. We now know that there exist true propositions which we can never formally prove. What about propositions whose proofs require arguments beyond our capabilities? What about propositions whose proofs require millions of pages? Or a million, million pages? Are there proofs that are possible, but beyond us? (Calvin Clawson, Mathematical Mysteries).

014 The multiplication symbol. In 1631, the multiplication symbol × was introduced by the English mathematician William Oughtred (1574-1660) in his book Keys to Mathematics, published in London. Incidentally, this Anglican minister is also famous for having invented the slide rule, which was used by generations of scientists and mathematicians. The slide rule’s doom in the mid-1970s, due to the pervasive influx of inexpensive pocket calculators, was rapid and unexpected.

015 Math and madness. Many mathematicians throughout history have had a trace of madness or have been eccentric. Here’s a relevant quotation on the subject by the British mathematician John Edensor Littlewood (1885-1977), who suffered from depression for most of his life: Mathematics is a dangerous profession; an appreciable proportion of us goes mad.

016 Mathematics and murder. What triple murderer was also a brilliant French mathematician who did his finest work while confined to a hospital for the criminally insane? (See Answer 1.3.)

017 Creativity and madness. There is a theory that creativity arises when individuals are out of sync with their environment. To put it simply, people who fit in with their communities have insufficient motivation to risk their psyches in creating something truly new, while those who are out of sync are driven by the constant need to prove their worth. They have less to lose and more to gain (Gary Taubes, Beyond the Soapsuds Universe, 1977).

018 Mathematicians and religion. Over the years, many of my readers have assumed that famous mathematicians are not religious. In actuality, a number of important mathematicians were quite religious. As an interesting exercise, I conducted an Internet survey in which I asked respondents to name important mathematicians who were also religious. Isaac Newton and Blaise Pascal were the most commonly cited religious mathematicians.

In many ways, the mathematical quest to understand infinity parallels mystical attempts to understand God. Both religion and mathematics struggle to express relationships between humans, the universe, and infinity. Both have arcane symbols and rituals, as well as impenetrable language. Both exercise the deep recesses of our minds and stimulate our imagination. Mathematicians, like priests, seek ideal, immutable, nonmaterial truths and then often venture to apply these truths in the real world. Are mathematics and religion the most powerful evidence of the inventive genius of the human race? In Reason and Faith, Eternally Bound (December 20, 2003, New York Times, B7), Edward Rothstein notes that faith was the inspiration for Newton and Kepler, as well as for numerous scientific and mathematical triumphs. The conviction that there is an order to things, that the mind can comprehend that order and that this order is not infinitely malleable, those scientific beliefs may include elements of faith.

In his Critique of Pure Reason, Immanuel Kant describes how the light dove, cleaving the air in her free flight and feeling its resistance against her wings, might imagine that its flight would be freer still in empty space. But if we were to remove the air, the bird would plummet. Is faith—or a cosmic sense of mystery—like the air that allows some seekers to soar? Whatever mathematical or scientific advances humans make, we will always continue to swim in a sea of mystery.

019 Pascal’s mystery. There is a God-shaped vacuum in every heart (Blaise Pascal, Pensées, 1670).

020 Leaving mathematics and approaching God. What famous French mathematician and teenage prodigy finally decided that religion was more to his liking and joined his sister in her convent, where he gave up mathematics and social life? (See Answer 1.4.)

021 Ramanujan′s gods. As mentioned in this book’s introduction, the mathematician Srinivasa Ramanujan (1887-1920) was an ardent follower of several Hindu deities. After receiving visions from these gods in the form of blood droplets, Ramanujan saw scrolls that contained very complicated mathematics. When he woke from his dreams, he set down on paper only a fraction of what the gods showed him.

Throughout history, creative geniuses have been open to dreams as a source of inspiration. Paul McCartney said that the melody for the famous Beatles’ song Yesterday, one of the most popular songs ever written, came to him in a dream. Apparently, the tune seemed so beautiful and haunting that for a while he was not certain it was original. The Danish physicist Niels Bohr conceived the model of an atom from a dream. Elias Howe received in a dream the image of the kind of needle design required for a lock-stitch sewing machine. René Descartes was able to advance his geometrical methods after flashes of insight that came in dreams. The dreams of Dmitry Mendeleyev, Friedrich August Kekulé, and Otto Loewi inspired scientific breakthroughs. It is not an exaggeration to suggest that many scientific and mathematical advances arose from the stuff of dreams.

022 Blaise Pascal (1623-1662), a Frenchman, was a geometer, a probabilist, a physicist, a philosopher, and a combinatorist. He was also deeply spiritual and a leader of the Jansenist sect, a Calvinistic quasi-Protestant group within the Catholic Church. He believed that it made sense to become a Christian. If the person dies, and there is no God, the person loses nothing. If there is a God, then the person has gained heaven, while skeptics lose everything in hell.

Legend has it that Pascal in his early childhood sought to prove the existence of God. Because Pascal could not simply command God to show Himself, he tried to prove the existence of a devil so that he could then infer the existence of God. He drew a pentagram on the ground, but the exercise scared him, and he ran away. Pascal said that this experience made him certain of God’s existence.

One evening in 1654, he had a two-hour mystical vision that he called a night of fire, in which he experienced fire and the God of Abraham, Isaac, and Jacob ... and of Jesus Christ. Pascal recorded his vision in his work Memorial. A scrap of paper containing the Memorial was found in the lining of his coat after his death, for he carried this reminder about with him always. The three lines of Memorial are

Complete submission to Jesus Christ and to my director.

Eternally in joy for a day’s exercise on the earth.

May I not forget your words. Amen.

023 Transcendence. Much of the history of science, like the history of religion, is a history of struggles driven by power and money. And yet, this is not the whole story. Genuine saints occasionally play an important role, both in religion and science. For many scientists, the reward for being a scientist is not the power and the money but the chance of catching a glimpse of the transcendent beauty of nature (Freeman Dyson, in the introduction to Nature’s Imagination).

024 The value of eccentricity. That so few now dare to be eccentric, marks the chief danger of our time (John Stuart Mill, nineteenth-century English philosopher).

025 Counting and the mind. I quickly toss a number of marbles onto a pillow. You may stare at them for an instant to determine how many marbles are on the pillow. Obviously, if I were to toss just two marbles, you could easily determine that two marbles sit on the pillow. What is the largest number of marbles you can quantify, at a glance, without having to individually count them? (See Answer 1.5.)

026 Circles. Why are there 360 degrees in a circle? (See Answer 1.6.)

027 The mystery of Ramanujan. After years of working through Ramanujan’s notebooks, the mathematician Bruce Berndt said, I still don’t understand it all. I may be able to prove it, but I don’t know where it comes from and where it fits into the rest of mathematics. The enigma of Ramanujan’s creative process is still covered by a curtain that has barely been drawn (Robert Kanigel, The Man Who Knew Infinity, 1991).

028 Calculating π. Which nineteenth-century British boarding school supervisor spent a significant portion of his life calculating π to 707 places and died a happy man, despite a sad error that was later found in his calculations? (See Answer 1.8.)

029 The world’s most forgettable license plate? Today, mathematics affects society in the funniest of ways. I once read an article about someone who claimed to have devised the most forgettable license plate, but the article did not divulge the secret sequence. What is the most forgettable license plate? Is it a random sequence of eight letters and numbers—for example, 6AZL4QO9 (the maximum allowed in New York)? Or perhaps a set of visually confusing numbers or letters—for example, MWNNMWWM? Or maybe a binary number like 01001100. What do you think? What would a mathematician think? (See Answer 1.7.)

030 The special number 7. In ancient days, the number 7 was thought of as just another way to signify many. Even in recent times, there have been tribes that used no numbers higher than 7.

In the 1880s, the German ethnologist Karl von Steinen described how certain South American Indian tribes had very few words for numbers. As a test, he repeatedly asked them to count ten grains of corn. They counted slowly but correctly to six, but when it came to the seventh grain and the eighth, they grew tense and uneasy, at first yawning and complaining of a headache, then finally avoided the question altogether or simply walked off. Perhaps seven means many in such common phrases as seven seas and seven deadly sins. (These interesting facts come from Adrian Room, The Guinness Book of Numbers, 1989.)

031 Carl Friedrich Gauss (1777-1855), a German, was a mathematician, an astronomer, and a physicist with a wide range of contributions. Like Ramanujan, after Gauss proved a theorem, he sometimes said that the insight did not come from painful effort but, so to speak, by the grace of God. He also once wrote, There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example, touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us and completely outside the province of science.

032 Isaac Newton (1642-1727), an Englishman, was a mathematician, a physicist, an astronomer, a coinventor of calculus, and famous for his law of gravitation. He was also the author of many books on biblical subjects, especially prophecy.

Perhaps less well known is the fact that Newton was a creationist who wanted to be known as much for his theological writings as for his scientific and mathematical texts. Newton believed in a Christian unity, as opposed to a trinity. He developed calculus as a means of describing motion, and perhaps for understanding the nature of God through a clearer understanding of nature and reality. He respected the Bible and accepted its account of Creation.

033 Genius and eccentricity. The amount of eccentricity in a society has been proportional to the amount of genius, material vigor and moral courage which it contains (John Stuart Mill, On Liberty, 1869).

034 Mathematics and God. The Christians know that the mathematical principles, according to which the corporeal world was to be created, are coeternal with God. Geometry has supplied God with the models for the creation of the world. Within the image of God it has passed into man, and was certainly not received within through the eyes (Johannes Kepler, The Harmony of the World, 1619).

035 James Hopwood Jeans (1877-1946) was an applied mathematician, a physicist, and an astronomer. He sometimes likened God to a mathematician and wrote in The Mysterious Universe (1930), From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician. He has also written, Physics tries to discover the pattern of events which controls the phenomena we observe. But we can never know what this pattern means or how it originates; and even if some superior intelligence were to tell us, we should find the explanation unintelligible (Physics and Philosophy, 1942).

036 Leonhard Euler (1707-1783) was a prolific Swiss mathematician and the son of a vicar. Legends tell of Leonhard Euler’s distress at being unable to mathematically prove the existence of God. Many mathematicians of his time considered mathematics a tool to decipher God’s design and codes. Although he was a devout Christian all his life, he could not find the enthusiasm for the study of theology, compared to that of mathematics. He was completely blind for the last seventeen years of his life, during which time he produced roughly half of his total output.

Euler is responsible for our common, modern-day use of many famous mathematical notations—for example, f(x) for a function, e for the base of natural logs, i for the square root of -1, π for pi, Σ for summation. He tested Pierre de Fermat’s conjecture that numbers of the form 2n + 1 were always prime if n is a power of 2. Euler verified this for n = 1, 2, 4, 8, and 16, and showed that the next case 2³² + 1 = 4,294,967, 297 = 641 × 6,700,417, and so is not prime.

037 George Boole (1815-1864), an Englishman, was a logician and an algebraist. Like Ramanujan and other mystical mathematicians, Boole had mystical experiences. David Noble, in his book The Religion of Technology, notes, The thought flashed upon him suddenly as he was walking across a field that his ambition in life was to explain the logic of human thought and to delve analytically into the spiritual aspects of man’s nature [through] the expression of logical relations in symbolic or algebraic form.... It is impossible to separate Boole’s religious beliefs from his mathematics.

Boole often spoke of his almost photographic memory, describing it as an arrangement of the mind for every fact and idea, which I can find at once, as if it were in a well-ordered set of drawers.

Boole died at age forty-nine, after his wife mistakenly thought that tossing buckets of water on him and his bed would cure his flu. Today, Boolean algebra has found wide applications in the design of computers.

038 The value of puzzles. It is a wholesome plan, in thinking about logic, to stock the mind with as many puzzles as possible, since these serve much the same purpose as is served by experiments in physical science (Bertrand Russell, Mind, 1905).

039 A mathematical nomad. What legendary mathematician, and one of the most prolific mathematicians in history, was so devoted to math that he lived as a nomad with no home and no job? Sexual contact revolted him; even an accidental touch by anyone made him feel uncomfortable. (See Answer 1.9.)

040 Marin Mersenne (1588- 1648) was another mathematician who was deeply religious. Mersenne, a Frenchman, was a theologian, a philosopher, a number theorist, a priest, and a monk. He argued that God’s majesty would not be diminished had He created just one world, instead of many, because the one world would be infinite in every part. His first publications were theological studies against atheism and skepticism.

Mersenne was fascinated by prime numbers (numbers like 7 that were divisible only by themselves and 1), and he tried to find a formula that he could use to find all primes. Although he did not find such a formula, his work on Mersenne numbers of the form 2p - 1, where p is a prime number, continues to interest us today. Mersenne numbers are the easiest type of number to prove prime, so they are usually the largest primes of which humanity is aware.

Mersenne himself found several prime numbers of the form 2p - 1, but he underestimated the future of computing power by stating that all eternity would not be sufficient to decide if a 15- or 20-digit number were prime. Unfortunately, the prime number values for p that make 2p - 1 a prime number seem to form no regular sequence. For example, the Mersenne number is prime when p = 2,